152 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 
The first thing that would occur to a systematist on 
looking at these lists is the extraordinary number of “ species.” 
There are 247 organisms in the first two lists, and these fall into 
51 categories. It would be safe to say that no one could 
collect, at random, 247 individual animals or plants belonging 
to (say) the same order, and find 51 separate species. Now 
in these identifications there are 8 reactions, each of which 
may be positive or negative, that is, 16 characters in all. 
At first the results seem to suggest that we are dealing with a 
series of combinations of the given characters, which are 
simply chance combinations, but this is not the case. We may 
compare the series of cases given in the tables with the cases 
that would arise on tossing 8 coins simultaneously. The first 
coin may fall in two ways, and each of these ways is to be 
associated with the two ways in which the second coin may fall, 
and so on, giving the number of possible results = 2 x 2 x 2, 
etc., or 28 = 256 combinations. But our first reaction, that 
with glucose, which ought to occur in two ways, only occurs 
in one way (the coin is “‘ loaded ” so that it always falls heads), 
and this is also the case with the reaction to lactose. The 
reaction to mulin was always negative (the coin always falls 
tails). The reaction to cane sugar occurs in the positive way 
far oftener than in the negative way (the coin is loaded so that 
it falls more often heads than tails), and just the opposite is the 
case with the V. and P. reaction. So likewise with the other 
reactions. The number of actual results is far less than the 
number of possible results. 
Looking at the frequencies of occurrence of the various 
results we see that these are 28, 26, 20, 12, 11, 11, 10, 7, 6, 6, 6, 
5,4, 4, 4, 3, 3, 3, 1, etc., where the first five terms represent> 
the relative frequencies of the organisms, McConkey’s No. 101, 
B. neapolitanus, B. vesiculosus, B. acidi lactics and B. lactis 
aerogenes, or we may look at the results in another way : 
one-third of all the organisms identified, and nearly one-half 
